Multistage Mathematics Instruction

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Multi-stage Mathematics Instruction

• Jeff Knisley
• East Tennessee State University
• MAA-Southeastern Section, Spring 2005

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I enjoy teaching

• Teaching is its own reward.
• The satisfaction of seeing students progress
• The joy of studying and sharing mathematics for a
  living
• The mutual benefit of the student-teacher
  relationship
• Favorite Quote They wont care what you know
  until they know that you care.

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But are they learning anything

• Performance doesnt always imply learning
• In the 70s, a student was taught MACSYMA but not
  Calculus
• Student Aced a series of MIT Calculus Tests
• I have often wondered
• Are they learning, or are they just good at
  taking my tests?
• How close are they to understanding a concept
  well enough to put it to good use?
• How can I make mathematics and problem solving
  less frustrating for them?

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How do students learn math?

• Early on, I felt I had to have an answer to this
  question in order to teach at all.
• Reviewed Math Education Research
• Explored Cognitive and Applied Psychology
• Had some experience in Artificial Intelligence
• I combined the research, some observations, and
  some simple experiments into a macro-model of
  how a student learns math

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Outline

• Mathematics Education, Applied Psychology
• What the Experts say about learning
• Some observations and simple experiments
• A Macro-model for mathematical learning
• As a guide for implementing new curricula
• Not an exact description of mathematics students
• Using the model to identify Best Practices
• As an indicator of what works and what doesnt
• As a guide for using Technology in mathematics

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What the Experts SayMath. Ed.

• Individual Learning Styles
• Some students are visual learners, some learn by
  synthesizing ideas, some learn by imitation
• Each student has a preferred learning style
• Preferred style used to construct concepts
• Kolb Learning Model
• Those who learn by building on previous
  experience
• Those who learn by trial and error
• Those who learn from detailed explanations
• Those who learn by implementing new ideas
• Much of this from R. Felder, Engineering
  Education

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What the Experts SayPsychology

• Learning Models
• Different people associate new ideas to old ones
  at different rates
• Different people memorize information at
  different rates
• Individual Differences in Skill Acquisition
• Give subjects simple air traffic control game
• First they discover simple heuristicshow to land
  planes, how to create holding patterns
• Their game abilities improve in jumps as they
  develop strategies that allow sophisticated
  actions

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What the Experts sayAI

• Heuristic reasoning
• Associates a pattern with an action
• Closest Pattern determines method used
• Criteria for closest often yields incorrect
  result
• Is knowledge without understanding
• Heuristics Rote Learning
• Reduces learning to a set of rules to memorize
• Replaces comprehension with association

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Example Heuristic Reasoning
Problem Simplify
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Observation Math Learning Types

• Kolb Learning styles translated into math
• Allegorizers They prefer form over function, and
  often ignore details
• Integrators They want to compare and contrast
  the known with the unknown.
• Analyzers They desire logical explanation and
  detailed descriptions
• Synthesizers They use known concepts like
  building blocks to construct new ideas.
• Other models yield similar Math Styles

Allegory (noun) figurative treatment of
one subject under the guise of another
(Webster)
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Experiment Pythagorean Theorem

• Procedure
• Present an example right triangle with sides
  given and hypotenuse unknown
• Prove the Pythagorean theorem and use it to
  determine the hypotenuse
• Measure the three sides of the triangle and show
  it satisfies Pythagorean theorem
• Distribute paper with right triangle with unknown
  hypotenuse ( and rulers)

?
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Expected Observations

• Allegorizers (_at_ 15 of sampled students)
• They reduce learning to a set of Case Studies
• They look in the text for a worked example
• Integrators (_at_ 60 of sampled students)
• Ruler is known, Hypotenuse is unknown
• They use the ruler to measure the hypotenuse
• Analyzers (_at_ 20 of sampled students)
• They use the Pythagorean theorem
• They seem to want a logical explanation
• Synthesizers (A handful, at best)
• Use theorem and explore to find a 3-4-5 triangle

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Observation Topic implies Style

• Style is a function of student and topic
• A student may be an analyzer in Linear Algebra
• Same student may be an allegorizer in Statistics
• We resort to Heuristics when all else fails
• Math Ed research shows that even the best
  students fail to understand limits
• Students pass tests on limits by resorting to
  heuristicsmemorization and pattern-based
  association

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Definition of the Macro-Model

• Students acquire new concepts by progressing
  through 4 stages of understanding
• Allegorization A new concept is described in
  terms of existing knowledge (i.e., intuitively)
• Integration Comparative analysis is used to
  distinguish new concept from known concepts
• Analysis New concept becomes part of existing
  knowledge. Connections and explanations follow.
• Synthesis New concept is used as a building
  block to establish new theories, new strategies,
  and new allegories

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The Importance of Allegories

• Learning begins with allegory development
• New concept stated in a familiar context
• Allegory is description within the given context
• Insufficient allegorization prevents learning
• Failure to Allegorize forces a Heuristic Approach
• Some good students have sophisticated heuristics

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Example Chess without Allegories
Valid moves for a given token are determined by
tokens type. Each player attempts to capture the
others F token.
Each player receives 8 A tokens, 2 each of
B, C, and D tokens, and 1 each of E and
F tokens
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Discussion Learning Chess

• Context is Medieval Military Figures
• Game pieces themselves are allegories
• Pawns are numerous but have limited abilities
• Knights can Leap over objects
• Queens have unlimited power
• Capture the King is the allegory for winning
• Colors are allegories
• White Versus Black
• Battles take place when 2 pieces occupy the
  same square on the checkerboard

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Components of Integration

• Student now understands allegorically that there
  is a new concept to be acquired
• Places a label on the new idea i.e., a
  definition
• Definition places concept into a mathematical
  setting
• Compare and Contrast
• How is new concept like known concepts?
• How does new concept differ from known concepts?
• We often neglect this stage
• Visual Comparisons are the most powerful
• Technology can be used to produce comparisons

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Tangent Planes
Which plane, A and B, is tangent to the surface
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Analysis of a New Concept

• The new concept takes on its own character
• Explanations and origins are developed
• Techniques for use of new concept are developed
• The new concept becomes one of many characters
• Connections to existing ideas are established
• Sphere of influence becomes well-defined
• What known concepts are related to new concept?
• How are known concepts modified by the new
  concept?
• Analysis desires that a great deal of relevant
  information be delivered quickly(i.e., lectures)

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Synthesis and Problem Solving

• New idea becomes a tool
• To create allegories for new ideas
• To create new versions of existing knowledge
• To solve problems and prove theorems
• Strategy development
• New concept and known concept are combined into
  sophisticated constructions
• New concept is used to solve problems
• Applications are desired and explored in depth

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Identifying Best Practices

• Too often, instruction is directed at analyzers
• Lectures and techniques
• Integrators and Allegorizers are lost/confused
• Synthesizers may get bored and fall behind,
  making them allegorizers for later material
• Most Students forced to use heuristics
• Integrators and Allegorizers memorize rules
• Analyzers often apply heuristics anyway
• Example Studies have proven this for Limits

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Example Uncertainty Principle

• Physics student asked me to explain Heisenberg
  Uncertainty to him from a Mathematical
  perspective
• Mathematical If A and B are self-adjoint and
• AB BA I,
• where I is the identity operator, and if f is
  in dom(A) n dom(B) vector with f 1, then
• Af Bf ½
• The amazing thing is that non-commutivity of A
  and B implies the lower bound

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Example Heisenberg Uncertainty

• Proof Define
• Q(t) (AitB)f 2
• Expand to show that
• Q(t) Af 2 t Bf 2 t2
• Q(t) 0 implies that
• 4 Af 2 Bf 2 2 1
• Main Example (Af)(x) xf(x), (Bf)(x) f '(x)
  over L2(R)

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Heisenberg Uncertainty

• Mathematically, Uncertainty is tricky
• f is differentiable a.e., but cannot allow f
  in dom(B) because Bf 0
• So how to explain the mathematics of uncertainty
  to a physics student?

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Using the Model

• Allegory relating derivative to multiplication
  operator
• Integration Interactive applet where they
  attempt to construct a function that minimizes
  uncertainty

Area under the curve 1
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Best Practices Defining Roles

• Role of the Teacher
• Allegorization Teacher is a story-teller
• Integration Teacher is a guide
• Analysis Teacher is an expert
• Synthesis Teacher is a coach
• Role of Technology
• Integration hands on student exploration
• Technology for integration
• After Introducing a concept
• Before extensive lecture on the concept

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Example Exponential Growth

• How to introduce the exponential (and later,
  logarithmic growth) to a biology student?
• Standard
• so there is 2
• The fact that yet satisfies y'y does not mean
  all that much to a biology student

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Using the Model

• Allegory Birth/Death processes
• A population growing at a rate of k per hour
  does not reproduce all at once.
• Instead, reproduction takes place many times per
  hour
• Exponential growth is a birth process with a
  constant rate in which reproduction takes place
  arbitrarily many times per hour

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Introductory Systems Ecology

• Integration
• Divide time interval 0,t into n short periods,
  where n is a very large integer
• Having n generations of reproduction means n
  periods where each period has length h t / n
• Probability of reproduction in each time period
  is kh, which is rate scaled over period
• Simulation Start with P individuals and let
  each reproduce over 1st period with probability
  kh. Repeat for all n time periods
  (http//faculty.etsu.edu/knisleyj/biomath/birthdea
  th.htm)

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Introductory Systems Ecology

• Start with P0 individuals
• After 1st period P1 P0 kh P0 P0 (1kh)
  individuals
• After 2nd period P2 P1 kh P1 P0 (1kh)2
  individuals
• After nth period Pn Pn-1 kh Pn-1 P0
  (1kh)n individuals
• n arbitrarily large means n approaching 8
• Definition The exponential function is defined
• and from this we can derive all properties of the
  exponential.

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Best PracticesTechnology

• Technology as intermediate assessment
• Multivariable Calculus All quizzes are Maple
  worksheets (http//faculty.etsu.edu/knisleyj/multi
  stage/quiz5.mw)
• Intro Stats
• 1200 students per semester in our gen ed course
• 4 stage instruction
• Lecture Applets Computer Assessment
• Lecture and Assessment are traditional
• Applets prepare for graded Minitab activities
• Technology for extended projects

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Technology as aid to Student Research

• Allegory Introduction to research problem,
  where problem is an extension of known result
• Integration -- Student uses Maple, NetLogo, etc.
  to reproduce or simulate known result
• Analysis Predict an answer to problem using
  technological extension of known result
• Synthesis Proof or otherwise solution of given
  research problem

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Research Examples

• M.P. began by reproducing a well-known
  agent-based army ant raiding pattern model en
  route to agent-based model of division of labor
  in social insects.
• P.C. began with simple implementation of classic
  Neural Net algorithm en route to using neural
  nets for data mining micro-array data
• A.T. began with simple curve-fitting algorithm en
  route to proving a version of the C.H. theorem
  for a class of elliptic operators

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Summary

• Allegory (intuition) that leads to integration,
  perhaps via technology and interactive
  assessment, so that they are ready for
  lecture-based analysis that fleshes out the
  concept, and then can begin to synthesize their
  own ideas
• But students cant create their own allegories or
  coach themselves as synthesizers. Thus, the
  model ultimately predicts the necessity of the
  mutually-beneficial student-teacher relationship.

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Thank you!

• Any Questions?